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Resolución de integrales/ (y-1)/ sqrt(1-(y-1)^2)

Integral de sqrt(1-(y-1)^2) dy

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  2                     
  /                     
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 |     ______________   
 |    /            2    
 |  \/  1 - (y - 1)   dy
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/                       
1
∫121−(y−1)2 dy\int\limits_{1}^{2} \sqrt{1 - \left(y - 1\right)^{2}}\, dy 
Integral(sqrt(1 - (y - 1)^2), (y, 1, 2))
Respuesta (Indefinida) [src] 
                              //                                   3                                                  \
                              ||  I*acosh(-1 + y)        I*(-1 + y)               I*(-1 + y)           |        2|    |
  /                           ||- --------------- + --------------------- - ---------------------  for |(-1 + y) | > 1|
 |                            ||         2               ________________        ________________                     |
 |    ______________          ||                        /              2        /              2                      |
 |   /            2           ||                    2*\/  -1 + (-1 + y)     2*\/  -1 + (-1 + y)                       |
 | \/  1 - (y - 1)   dy = C + |<                                                                                      |
 |                            ||                             _______________                                          |
/                             ||                            /             2                                           |
                              ||           asin(-1 + y)   \/  1 - (-1 + y)  *(-1 + y)                                 |
                              ||           ------------ + ---------------------------                   otherwise     |
                              ||                2                      2                                              |
                              \\                                                                                      /
∫1−(y−1)2 dy=C+{i(y−1)32(y−1)2−1−i(y−1)2(y−1)2−1−iacosh⁡(y−1)2for ∣(y−1)2∣>11−(y−1)2(y−1)2+asin⁡(y−1)2otherwise\int \sqrt{1 - \left(y - 1\right)^{2}}\, dy = C + \begin{cases} \frac{i \left(y - 1\right)^{3}}{2 \sqrt{\left(y - 1\right)^{2} - 1}} - \frac{i \left(y - 1\right)}{2 \sqrt{\left(y - 1\right)^{2} - 1}} - \frac{i \operatorname{acosh}{\left(y - 1 \right)}}{2} & \text{for}\: \left|{\left(y - 1\right)^{2}}\right| > 1 \\\frac{\sqrt{1 - \left(y - 1\right)^{2}} \left(y - 1\right)}{2} + \frac{\operatorname{asin}{\left(y - 1 \right)}}{2} & \text{otherwise} \end{cases}
Simplificar
Gráfica
1.002.001.101.201.301.401.501.601.701.801.9002
Trazar el gráfico f(x)
Respuesta [src] 
  2                                                                                                                     
  /                                                                                                                     
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 |  /                                        2                  3                                                       
 |  |           I                3*I*(-1 + y)         I*(-1 + y) *(1 - y)      I*(1 - y)*(-1 + y)               2       
 |  |- ------------------- + --------------------- + --------------------- - ---------------------  for (-1 + y)  > 1   
 |  |     ________________        ________________                     3/2                     3/2                      
 |  |    /              2        /              2      /             2\        /             2\                         
 |  |  \/  -1 + (-1 + y)     2*\/  -1 + (-1 + y)     2*\-1 + (-1 + y) /      2*\-1 + (-1 + y) /                         
 |  |                                                                                                                   
 |  <                 _______________                                                                                 dy
 |  |                /             2                                                                                    
 |  |              \/  1 - (-1 + y)              1               (1 - y)*(-1 + y)                                       
 |  |              ------------------ + -------------------- + --------------------                     otherwise       
 |  |                      2                 _______________        _______________                                     
 |  |                                       /             2        /             2                                      
 |  |                                   2*\/  1 - (-1 + y)     2*\/  1 - (-1 + y)                                       
 |  \                                                                                                                   
 |                                                                                                                      
/                                                                                                                       
1
∫12{i(1−y)(y−1)32((y−1)2−1)32−i(1−y)(y−1)2((y−1)2−1)32+3i(y−1)22(y−1)2−1−i(y−1)2−1for (y−1)2>1(1−y)(y−1)21−(y−1)2+1−(y−1)22+121−(y−1)2otherwise dy\int\limits_{1}^{2} \begin{cases} \frac{i \left(1 - y\right) \left(y - 1\right)^{3}}{2 \left(\left(y - 1\right)^{2} - 1\right)^{\frac{3}{2}}} - \frac{i \left(1 - y\right) \left(y - 1\right)}{2 \left(\left(y - 1\right)^{2} - 1\right)^{\frac{3}{2}}} + \frac{3 i \left(y - 1\right)^{2}}{2 \sqrt{\left(y - 1\right)^{2} - 1}} - \frac{i}{\sqrt{\left(y - 1\right)^{2} - 1}} & \text{for}\: \left(y - 1\right)^{2} > 1 \\\frac{\left(1 - y\right) \left(y - 1\right)}{2 \sqrt{1 - \left(y - 1\right)^{2}}} + \frac{\sqrt{1 - \left(y - 1\right)^{2}}}{2} + \frac{1}{2 \sqrt{1 - \left(y - 1\right)^{2}}} & \text{otherwise} \end{cases}\, dy 
=
= 
  2                                                                                                                     
  /                                                                                                                     
 |                                                                                                                      
 |  /                                        2                  3                                                       
 |  |           I                3*I*(-1 + y)         I*(-1 + y) *(1 - y)      I*(1 - y)*(-1 + y)               2       
 |  |- ------------------- + --------------------- + --------------------- - ---------------------  for (-1 + y)  > 1   
 |  |     ________________        ________________                     3/2                     3/2                      
 |  |    /              2        /              2      /             2\        /             2\                         
 |  |  \/  -1 + (-1 + y)     2*\/  -1 + (-1 + y)     2*\-1 + (-1 + y) /      2*\-1 + (-1 + y) /                         
 |  |                                                                                                                   
 |  <                 _______________                                                                                 dy
 |  |                /             2                                                                                    
 |  |              \/  1 - (-1 + y)              1               (1 - y)*(-1 + y)                                       
 |  |              ------------------ + -------------------- + --------------------                     otherwise       
 |  |                      2                 _______________        _______________                                     
 |  |                                       /             2        /             2                                      
 |  |                                   2*\/  1 - (-1 + y)     2*\/  1 - (-1 + y)                                       
 |  \                                                                                                                   
 |                                                                                                                      
/                                                                                                                       
1
∫12{i(1−y)(y−1)32((y−1)2−1)32−i(1−y)(y−1)2((y−1)2−1)32+3i(y−1)22(y−1)2−1−i(y−1)2−1for (y−1)2>1(1−y)(y−1)21−(y−1)2+1−(y−1)22+121−(y−1)2otherwise dy\int\limits_{1}^{2} \begin{cases} \frac{i \left(1 - y\right) \left(y - 1\right)^{3}}{2 \left(\left(y - 1\right)^{2} - 1\right)^{\frac{3}{2}}} - \frac{i \left(1 - y\right) \left(y - 1\right)}{2 \left(\left(y - 1\right)^{2} - 1\right)^{\frac{3}{2}}} + \frac{3 i \left(y - 1\right)^{2}}{2 \sqrt{\left(y - 1\right)^{2} - 1}} - \frac{i}{\sqrt{\left(y - 1\right)^{2} - 1}} & \text{for}\: \left(y - 1\right)^{2} > 1 \\\frac{\left(1 - y\right) \left(y - 1\right)}{2 \sqrt{1 - \left(y - 1\right)^{2}}} + \frac{\sqrt{1 - \left(y - 1\right)^{2}}}{2} + \frac{1}{2 \sqrt{1 - \left(y - 1\right)^{2}}} & \text{otherwise} \end{cases}\, dy 
Integral(Piecewise((-i/sqrt(-1 + (-1 + y)^2) + 3*i*(-1 + y)^2/(2*sqrt(-1 + (-1 + y)^2)) + i*(-1 + y)^3*(1 - y)/(2*(-1 + (-1 + y)^2)^(3/2)) - i*(1 - y)*(-1 + y)/(2*(-1 + (-1 + y)^2)^(3/2)), (-1 + y)^2 > 1), (sqrt(1 - (-1 + y)^2)/2 + 1/(2*sqrt(1 - (-1 + y)^2)) + (1 - y)*(-1 + y)/(2*sqrt(1 - (-1 + y)^2)), True)), (y, 1, 2))
Respuesta numérica [src] 
0.785398163397448
0.785398163397448

    Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.

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